DTIC ADA031961: The Limit-Point, Limit-Circle Nature of Rapidly Oscillating Potentials.

Au-AuJl 961

UNCLASSIFIED

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WISCONSIN UNI V MAO I SON MATHEMATICS RESEARCH CENTER F/G 12/1

the limit-point# limit-circle nature OF RAPIDLY oscillating POT— ETC IU)
SEP 76 F V ATKINSON# M S EASThAM# J B MCLEOO QAAG29-75-C-0024
MRC-TSR-1676 nL

AD A 0 3 1 961

V

Mathematics Research Center
University of Wisconsin— Madison

610 Walnut Street
Madison, Wisconsin 53706

September 197 6

Received September 25, 197 5)

Approved far public release
Distribution unlimited

Sponsored by

U.S. Army Research Office
P. O. Box 12211
Research Triangle Park
North Carolina 27709

c

UNIVERSITY OF WISCONSIN - MADISON
MATHEMATICS RESEARCH CENTER

THE LIMIT-POINT, LIMIT-CIRCLE NATURE OF RAPIDLY
OSCILLATING POTENTIALS

F. V. Atkinson, M. S. P. Eastham, and J. B. McLeod

Technical Summary Report # 167 6
September 1976

ABSTRACT

The report analyses the Weyl limit-point, limit-circle classification,

2

i.e. the number of linearly independent solutions in L (0,«), of the
equation

y"(x) - q(x)y(x) = 0 (0 < x < ») ,

where q(x) has the form

q(x) = x°p(x|3) ,

a and (3 being positive constants and p( t) a real continuous periodic
function of t.

AMS (MOS) Subject Classification: 34B20

Key Words: Ordinary differential operators

r ICf.'iiW ,5f

1 MR

Boundary-value problems

1 c:c

Self-adjointness

| « as ■■n

\ J'.illl .i*™*

Asymptotics

Work Unit Number l (Applied Analysis)

1,1

lAl

.i

Sponsored by the United States Army under Contract No. DAAG29-7 5-C-0024.

□ O

THE LIMIT-POINT, LIMIT-CIRCLE NATURE OF RAPIDLY

k *

OSCILLATING POTENTIALS

T. V. Atkinson, M. S. P. Eastham, and J. B. McLeod
1 . Introduction

We consider the Weyl limit-point, limit-circle classification, i.e. the
number of linearly independent solutions in L^(0, «>), of the second-order equation

y"(x) - q(x)y(x) = 0 (0 < x < «) , (l.l)

where the real-valued potential q(x) has the form

q(x) = xap(xti) . (1.2)

Here a and (3 are positive constants and p(t) is a continuous
periodic function of t. We denote by a the period of p(t).

It is perhaps worth remarking briefly on the significance of the
classification into limit-point and limit-circle for general real-valued
potentials q(x). In the limit-point case, the linear operator

- + q(x)

dx

t

associated with the equation (1.1) and some homogeneous boundary condition
at x = 0, say y(0) = 0, is self-adjoint (and so enjoys a well-defined
spectral theory) without the need to impose any boundary condition at <*.

In the limit-circle case, on the other hand, the operator does not become

Sponsored by the United States Army under Contract No. DAAG29-7 5-C-0024.

self-adjoint until its domain is restricted by the imposition of some
suitable boundary condition at «, and for each of these boundary
conditions there is a different spectrum. From the quantum-mechanical
point of view, where we expect a well-defined spectrum without the need
to impose additional boundary conditions, the limit-point case is the
more natural, but a discussion of this and some related topics is given in [10].

If we turn now to the particular case of (1. 2), one simple remark can be

made at the outset and this is that, if a < 2, (1. 2) makes (1.1) limit-point

for all |3. This follows from the well-known Levinson limit-point criterion
2

q(x) > -kx [4, p. 2 31], k a positive constant, which is applicable if
a < 2 because p(t), being periodic, is bounded below. The situation is
less simple if a > 2 and the object of this paper is to analyse the limit-
point, limit-circle nature of (1.2) for all a and p. In view of the simple

remark made above, we assume from now on that a > 2.

A partial analysis of two particular cases of (1.2) has already
appeared in the literature. The first case is p(t) = sin t, for which
(1. 2) was shown by Eastham [ 5] (see also [12]) to be limit-point if
p < 2. The range p < 1 had previously been covered by the work of
Hartman [11] and McLeod [15-17]. The second case is p(t) = -1 + k sin t,
where k is a constant. This time (1.2) was shown by Eastham [6] to

be limit-circle if p > ~a + \ and to be limit-point if p < 2 and I k J > 1

O 4

(see also [7]). Some corresponding results for fourth-order differential
equations have been given recently by Atkinson [2] and Eastham [9].

-2-

Throughout the paper, we denote by M the mean value of p(t) over (Q,a), i.e.

-1 r

M = a / p(t)dt .

0

In the paragraphs which follow, we divide our analysis of (1. 2) into various
cases. In the range a < 2(3 - 2, the results depend on whether M = 0,

M > 0, or M < 0. In the range a > 2(3 - 2, the results depend on
whether p(t) takes a positive value or not. These results are summarised
on the accompanying figure. The situation on the line a = 2p - 2 is
a special one and is described in §9 below. It will be seen from the
figure that our analysis is complete as far as the regions a < 2 and
a < 2(3 - 2 are concerned. For the region in which a > 2 and a > 2(3 - 2
our analysis is incomplete in that

(i) when p(t) <0 everywhere, differentiability conditions are
imposed on p(t) (see §7 below for a more detailed statement of these
conditions);

(ii) the case where p(t) < 0 but p(t) ^ 0 everywhere is not fully
dealt with. The situation seems to depend not only on a and (i but also
on the order of the zeros of p(t). The information that we have on this
case is given in §8 below.

-3-

"TV"

2. The case M = 0, u < p

We define

t

P(t) = f p(u)du .

0

Then the condition M - 0 implies that P(t) has period a and hence

that P(t) is bounded for all t. We refer now to a particular case of a

limit-point criterion of Brinck [3), that (1.1) is limit-point if

J x ^q(x)dx > - C (2.1)

I

for ail intervals J in, say, (l,00) of length <1, where C is a
constant, in our case of (1.2), we have

I x *q(x)dx = fi * f xa Pp(xP)d(xP)

J J

= p‘1[x°''PP(xP)] - p_1(a - P) / x°'P_1P(xP)dx .

1 J

Since P(xp) is bounded for all x > 0 and since we are assuming in

this section that a < (3, we have

I f x *q(x)dx | < C
J

for some constant C, and so (2.1) is certainly satisfied.

That oscillating potentials of the kind considered here might be
covered by (2.1) was suggested by Brinck himself [ 3, p.229] and he gave
the example q(x) = x“sin(xa+V

The result, then, of this section is:

4. 1&1 M = 0 and let « < p. Then (1.2) makes (1.1) limit-point-

-5-

We remark that A can also be proved by means of a limit-point
criterion which is of the same kind as the one in [ 5 | and can even be
deduced from it - that (1.1) is limit-point if there is a sequence of non-
overlapping intervals (a , b ) in [ 0, >») with ( b - a )* - «•>

nr m ’ m m

and such that

(bm ~ am) f q(x)dx > - C (2.2)

for all intervals I C (a b ). This criterion is given specifically in [ 2 I

mm 1

as a particular case of results for higher-order differential equations.

It is also of the same nature as the criterion in [ 3). The choice to be

i i i _i

made in our case of (1. 2) is a = m2, b = m2 + - m

m m 4

-6-

m

3. The case M = 0, (3 < a < 2(3 - 2

We note that, since a > 2, the condition a < 2(3 - 2 implies
that (3 > 2. Hence the stated range [3 < a < 2(3 - 2 is meaningful.

We transform (1.1 - 2) to a more manageable form by means of the
transformation of Liouville type

t = xP, z(t) = x(P_1)/2y(x) . (3.1)

Then we obtain

z(t) + { bt 2 - (3 2t 2Vp(t)}z(t) = 0 , (3.2)

where b = ■“ (1 - (3 2) and

2\ = 2 - (a + 2)/(3 . (3. 3)

In this section, we determine the asymptotic form of the solutions
of (3. 2) as t -► «. Our method requires that

0 < 2y < 1 , (3.4)

i.e. , by (3. 3),

(3 - 2 < a < 2£ - 2 , (3. 5)

and this is certainly ensured by the stated range (3 < a < 2(3 - 2 .

In ( 3. 2) we substitute

z(t) = u(t)v(t) , (3. 6)

where

v(t) = tY{l + ^ P(t)t"2nY + r(t)t"2Y_1} . (3.7)

1

-7-

✓

Here the integer N is chosen to make

(2N 4 1)\ > 3y + 1 (3. 8)

and the p^(t) and r(t) are twice continuously differentiable periodic

functions, with period a, which are defined below.

With the substitution (3.6), (3.2) becomes

2d 2 du 3r •• -2 -2 -2\ , . , . ,

v ~ v — + uv [ v + {bt - |3 t p(t)}v] = 0 . (3.9)

Our intention is that the coefficient of u in (3.9) should approach a
positive constant as t -► °o.

3

Now v has the form

3N

V3 = t3v{l 4 £ r (t)t_2nv + 0(t“2v_1)} , (3.10)

1 n

where r^t) has period a, and r^t) does not involve p^^j(t), . . . , p^(t) .

In particular, we note that

rj(t) = 3pj(t) . (3.11)

Also,

N

v+{bt 2-f3 2t 2^p(t)}v=tV^ t'2nY(pn(t) - p"2p(t)pn_1(t)} + (b 4 v( V ~ 1) }tY_ 2 -

- 2 y t~ V_1pA(t) 4 t"Y_1r(t) 4 0(t"3Y_1) 4 0(t'(2N+1)Y) ,

where pQ(t) = 1 and the O-terms refer to t — ». By (3.8), the second
O-term can be neglected. Hence, using also (3.10), the coefficient of
u in (3.9) has the form

-8-

(3.12)

t3VV t’Zn^{p (t) + s (t)} + {b + v(v - l))tV 1 -
j n n

-v-1. - v- 1 .. _3v_i

- 2Y t p^t) + t r (t) + 0(t )) ,

where sn( t) involves, besides p(t), at most those p^(t) and r(t)
with j < n - 1 and hence at most the p.(t) with j < n - 1. We note
in particular that

s^t) = - p 2p(t)

and

s2(t) = - p 2p(t)p1(t) + {pj(t) - p 2p(t)}r1(t) .

Let M denote the mean value of s (t) over (0,a). Then the
n n

periodic functions p (t) are defined for n = 1, 2, . . . , N in turn by

n

p (t) = - s (t) + M .
n n n

Also, the periodic function r(t) is defined by

r(t) = 2\ p^t) .

We note that, by (3.13),

M = - p~2M = 0 .

Also, since p^t) = - s (t) = p 2p(t), again by ( 3. 13), (3.14) gives

S2U) = " P^PjU) •

Hence

a a

aM2 = / s2(t)dt = - [ p1(t)p1( t) ] q + J Pj (t)dt .

-9-

3.13)

3.14)

3.15)

3.16)

3.17)

Thus M. >0 and we write M = A . We now substitute (3.1^) and
2

(3.16) into (3.12). Then (3.12) takes the form

AZ + R(t) + {b + \(\ - l)}^”2 + 0(t'2) ,

(3.18)

where

m - 1 m/12"-4’''

3

3.19)

In (3.9), we now make the change of variable

£ = f v ^(t)dt = (1 - 2\) ^{1 + 0(t T)} ,

0

_i 1

except that the O-term would be 0(t * log t) if 2\ = ^ • Then, writing

u(t) = U(£), R(t) = Rj(|)
and using (3.18), we can write (3.9) as

.-2\v

(3.20)

~T + U(A2 + R (£) + 0(fd)) = 0 ,
d£

3. 21)

where

d = min{2,(l - 2\) *} > 1 • (3.22)

We note that, by (3. 4) and (3. 20), £ - 00 as t - * .

By (3.19), we have

S

dR,(£)

d£

d£ = /

dRUl

dt

dt < *> ,

and hence the asymptotic form of the solutions of (3. 21) as £ -* 00 follows
from the remarks on pp. 9l-9 2 of l 4 j . Thus (3.21) has two solutions

-10-

which are asymptotic respectively to

exp(± i f {A + R,(£) } ad|) .

0

In particular, going back through (3.7), (3.6), and (3.1), we find that
(1.1) has two solutions y^(x), (j = 1,2) such that

| y. (x) I ~ x

(3V-((3-l)/2

i.e. , by (3. 3),

I y j (x) I ~ x.

(p-a-l)/2

as x - ao. If a > (3, these two solutions are both L (O.oo)^ and

so we have the limit-circle case for (1.1). Thus the result of this section is:

B. Let M = 0 and let (3<«<2|3-2. Then (1.2) makes (1.1)
limit-circle.

We pointed out in (3. 4 - 5) that the above method up to (3. 23)
works when (3 - 2 < a < 2(3 - 2. Therefore it also follows from (3.2 3)
that, when (3 - 2 < a < (3, (1-2) makes (1.1) limit-point and, to this

extent, we have an overlap with the result A of §2.

We conclude this section by mentioning, first, that our method
has some points of similarity with the one indicated in § §2 and 5 of [IB]
and, secondly, that a possible alternative method would be to compare
(3.2) with the periodic equation z(t) - t p(t)z(t) = 0, where t is a
small parameter (cf. [ 1 ] ) .

» ii "fttlii mdii nuHTjur

f

4. The case M > 0, a < 2p - 2

Suppose first that a < p. Then we can use (2.1) again because now
I x *q(x)dx = f Mx“ *dx + ( x ^+Lt{p(x^) - M }dx

J J J

> f x *+°{p(x^) - M}dx ,

J

and this integral is bounded as in §2 since p(t) - M has mean value
zero. Thus the limit-point case occurs.

Now suppose that p < a < 2p - 2. Here we need only take the
case N = 1 of (3.7) and omit the term involving r(t). Thus we define

v(t) = tY{l 4 Pl(t)t“ 2v) ,
where p^(t) is the periodic function defined by

Pjft) = p 2{p(t) - M} . (4.1)

Then

v + { bt 2 - p 2t 2vp(t)}v

= y(y-l)t^ 2 + P *"t Y{p(t) - M } + btV 2-p 2t ^p(t)+o(t V) = -Mp 2t Y+o(t Y)
since \ - 2 < - \. Hence (3.9) is

v2^v2f-u(Mp-2t2Yt^)} = °.

Since {• • • } here is large and positive for large t, this equation has

c 2

solutions which are exponentially (and more) large in £ = | dt/v . Thus

0

1-2 V

we have a solution u(t) such that u(t) > exp(k t ), where k > 0.

2

Then certainly the corresponding solution y(x) of (1.1) is not L ( 0 , °°) ,
and again the limit-point case occurs. Thus the result of this section is:

C. Let M > 0 and let a < 2p - 2. Then (1.2) makes (1.1) limit-point.

-12-

f

5. The case M < 0, a < 2(3 - 2

7 5

Suppose first that p > — a + ~ . Then the situation is covered by

o 4

the analysis in § § 3 - 4 of [ 6 ] - see especially (4.4) and (4.5) of [ 6] .
We again define p^(t) by (4.1) above and then define

s(x) = x° 2|^Zp1(xfi)

Hence (1. 2) can be written

q(x) = Mx° + s"(x) + 0(xa ^)

and s(x) satisfies the conditions on S in Theorem 2 (and its modification)
in (6). Then, as in ( 6), we have the limit-circle case.

Suppose next that (3-2<a<2(3-2. Thus (3. 5) holds and we
shall consider again the method of §3. We note first, however, that
these two sub-cases (3 > ^ a + ~ and (3-2<a<2(3-2 overlap and
between them make up the whole of a < 2p - 2, subject of course to
the condition a > 2 which is assumed throughout.

We make the substitution (3.6 - 7) again. The condition M = 0
which was imposed in §3 was not in fact used until (3.17). If now M * 0,
(3.18) is replaced by

s(t) + {b + y(y - i)h4v'2 + o(t-1) ,

where

N /,

S(t) = l

1 * *

-1 3-

HMMW

a-:;'- i. -v ,»•«■»*

and, as in (3.17), M = - p 2M(* 0 now). Thus the leading term in

S(t) is

-2 2v

- 0 Mt y , (5.1)

which is large and positive as t — <*>. Correspondingly, (3.21) is
replaced by

+ u{s (4) + o(fd)} = o ,
dr

where S^(|) = S(t).

We now substitute

u(|) = Sj^UM!) (5.2)

and write

*1 = / s[/2(|)d4 • (5.3)

Then we obtain, as we did ( 3. 9) and ( 3. 21),

+ W(1 * S:3/4(£) s'1/4(|) + 0(£'dS'‘(£))) = 0 , (5.4)

dn 1 d|2 1 1

where W( r)) = w(£). By (5.1) and (3.20), the coefficient of W here is

1 + 0(t_1) + 0(t”d(1"2v^"2Y) . (5.5)

Since 2y < 1 and d > 1, by ( 3. 4) and (3. 22), we have

- d(l - 2Y) - 2Y < - 1 ,

this inequality being a re-arrangement of (d - 1)(1 - 2Y) > 0. Hence
(5. 5) is

l + o(t"1) = l + o(n”1/(1~Y))

-14-

a 'a

rz

H by (5. 3) and (3. 20). Since 1/(1 - y) > 1, we can again quote pp. 91 - 92

of [ 4) to say that all solutions W(^) of (5.4) are bounded as q - «.

Hence, going back through ( 5. 2), (5.1), (3.7), (3.6), and (3.1), we find
that all solutions y(x) of (1.1) are

o^evIP-D)/*, o(x”a/4)

as x -*«o. Since a > 2, all solutions of (1.1) are, therefore, LZ(0, «)
and we have the limit-circle case. Thus the result of this section is:

D. Let M < 0 and let a < 2(3-2. Then (1.2) makes (1.1) limit-circle.

6. The case u > 2[i - 2, pit) taking positive values

In (3. 3), we now have y < 0 and we write y = - 6, so that
6 > 0. Then in (3.2) we write

F(t) = fTV'Sp(t) - bfZ ,

so that

z(t) = F(t)z( t) .

(6.1)

Since p(t) is a periodic function which is now assumed to take positive

2 2

values, we can say that there are positive constants A , B , 0^, 6^
such that

.2 26 . 2 26
A t < F(t) < B t

(6.2)

in the intervals (0^ + na, 0^, + na) , n * 0,1,... . We call these

intervals (an, b^). By taking 0^ - 0^ small enough, we can arrange that

B<|a. (6.3)

Let z ,(t) and z At) be the solutions of (6.1) defined by
n, 1 n, 2

Zn = °» Zn |(dJ = 1 *

n, i n n, 1 n

z (t) = z (t) f z 2 (u)du .
n,2 n, 1 ' n, 1

(6.4)

(6.5)

We note that

W<zn,l'2n,2,(t' = •* '

(6.6)

Let N be any integer. From (6. 2), we have for n > N

4 s r<t> ^

(6.7)

16-

mm

in (a ,b ), where
n n ’

H - Aa and K = (B/A)sup(b /a .
n n n n

n>N

By (6. 3), we can choose N so that

*<!■

By (6-7), the theory of differential inequalities [20, p. 69 | applied
to (6.1) and (6.4) gives

a lsinh{u (t - a )} < z .(t) < (Ku ) ^inhfKu (t - a )} .
n 11 n n, i n n n

Then (6 . 5) gives

sinh{Kp (t - a )}sinh{u. (b - t)}
n n n n

"n, 2V L/ ~ K sinh{p (t - a )}sinh{u (b - a )}
rn n *n n n

z_ M) <

and

K sinh{pn(t - a^) }sinh{K^n(bn - t) }

'n, 2' L/ — sinh{Kp (t - a )}sinh{Ku. (b - a )}
n n n n

z_ ,(t)>

Now consider any two real solutions z^t), z2(t) of (6.1) such that

W(Zj, z^)(t) = 1 .

In (a , b ), we must have
n’ n ’

z (t) = A z (t) + B z (t) ,
1 n n, 1 n n, 2

z (t) = C z (t) + D z (t) ,
2 n n, 1 n n, 2

and (6. 6) and (6. 12) imply that

AD - B C = - 1 .
n n n n

(6.8)

(6.9)

(6.10)

(6.11)

(6.12)

(6.13)

-17-

It follows from (3.1) that (1.1) will be limit-point if, for all intervals

(a^, b^) with n large enough,

bn b

either f z2(t)t 2+2/^dt > k or f z2(t)t~2 + 2/Pdt > k ,

a a

n n

k being a positive constant independent of n. Now,

f zf(t)t 2 + 2/|3dt = A2I + 2A B J + B2I _ , (6.15)

1 n n, 1 n n n n n, 2 ’ ' '

where

n, i J n, 1 n n ’

'2t2/|Jdt < k .T2 2(K'1),‘n(bn'an)

Jn = / zn .(t)z 2(t)t"2 /Pdt < kn"2a“2+2/|ie

n "i n, 1 n, 2 rn n

a

n

on using (6.9), (6.10), and (6.11). By completing the square in (6.15),

we see that

/ . zf U)t

f(t)t-2+2/Pdt > A2(! T -J2)/I
1 - nv n, 1 n, 2 n n, 2

Then (6.14) certainly follows for Zj(t) if

A2 >

n - ^n n

(6.16)

where, in neglecting J in comparison to I .1 _, we have used

n n, i n, z

the inequality 4(K - 1) < 2, which is implied by (6.8). Similarly,
(6.14) follows for z^(t) if

y -2p (b -a )

_ 2 , 3 2-2/p n n n

C >ku a re
n — ^n n

If neither (6. 16) nor (6. 17) holds, then, by (6.13), we must have

either

2 -3 -2+2/p ^n n n

B > ku a e

n — n n

2 -3 -2+2/p 2tin(bn"an)

D > ku. a e

n — rn n

and then the inequality

/ z2(t)t"2+2/,3dt > B2(I I - J2)/I ,

1 n n, 1 n, <- n n, 1

obtained again from (6.15) by completing the square, gives (6.14) for
Zj(t) in the case of (6.18). We can argue similarly for z^( t) in the case
of (6.19). Hence the result of this section is:

E. Let o > 2p - 2

p(t) take positive values. Then

(1. 2) makes (1.1) limit-point.

We remark that the result E for the more restricted range a > 4p - 6
follows from a general limit-point criterion of Ismagilov [ 1 3) - see also 1 1 4 J .

-19-

7 . The case a > 2^ - 2, p(t) < 0 for all t

As in §6, we write y - - 6 in (3.2) and, since p(t) < 0 now,
-2 -4

we can write (3 p(t) = - Q (t), where Q(t) > 0. Then (3.2) is

z(t) + {bf2 + t26Q"4(t)}z(t) = 0 .

We make the substitution (3.6) again:

z(t) = u(t)v(t)

but, instead of (3.7), we take

v(t) = fi6Q(t){l + V pn(t)t'2n6} ,

1

where the integer N is chosen to make

2(N + 1)6 > 6 + 1

and the pn(t), to be defined below, have period a. Then, as for (3.9),

v2Av2S + Ut'^Q3(t)U + f Pn(t)t‘2nt)3x

x t 2 6(t){l + £ p (t)t-2n’} + 2t Q(t) £ Pn(t)t'2nJ +
1 1 n

-76 N , . -76-1 \

+ t Q(t) £ Pn(t)t'2n6 + 0(t )J +

+ {bt‘2 + t26Q‘4(t)}t‘26Q4(t){l + J p (t)t'2n6)4 = 0 ,

f n

jL

which is

2d 2 du
V dt V dt ' U

t^Vdtu * v pm,-2"6}3*

1

/

N

Q(t) {1+2; PJ1)*

1

V , \ -2n6->

> r\ ( + J

N

+ 2Q(t) £ Pn(t)t"2n° +
1 n

N

Q(t) Yj Pn(*)t
1

-2n6

N

+ U + E Pn(t)t

1

-2n6} +o(r26-1)

= 0 .

4

On expanding {• • • } here by the binomial theorem, we obtain (inter
alia) the terms 4pn(t)t ° (n = 1 N). Then pn(t) is chosen

so that there is no term involving t

-2n6

in [•••]. Thus

(7.5)

Pj(t) = - ^ Q3(t)Q(t)

and P (t) (n > 2) involves p(t),...,p (t). Hence the p (t) are

n 1 n-i n

determined in turn.

It is clear also that p^(t) involves Q^2N\t) and so we must
assume the existence and continuity of 0^2N + 2^(t) and hence of
p(2N The nearer 5 is tQ zero> the larger N is (by (7.4)) and

the greater the differentiability required of p(t). If 6 > 1 (for example),
i.e. if a > 4p - 2, we can take N = 0 in (7.4) and then we need
only assume the existence and continuity of p(t). More generally, if

-21-

••n* *

p(2N+2)^ exists and is continuous, we can deal with the region
a > 2(3 7^ ~~ - 2 since this last inequality is just a re-arrangement
of (7. 4). If, therefore, we wish to deal with the entire region a > 2p - 2
with a single differentiability condition on p(t), that condition has
to be that p(t) is infinitely differentiable.

With our choice of the p (t) described above, (7. 5) takes the

n

form

2d 2 du _ -2(N+1)6, _. -26-1. ,

v — v — + uU + 0(t ) + 0(t )] = 0 .

(7.6)

We make the change of variable

t _?

I = / v (t)dt .

0

By (7. 3), i/t ^ lies between positive constants as t -* °°. Hence
(7.6) becomes

2

^“7 + U{1 + 0(1" d)} = 0 , (7.7)

dr

where U(|) = u(t), d > 1, and we have used (7.4).

\

Since all solutions U(£) of (7.7) are bounded as | - », again

by pp. 91 - 9 2 of [ 4 ] , it follows from (7.2) and (7.3) that all solutions

--6

z(t) of (3.2) are 0(t 2 ) as t -* ». Hence, by (3.1), all solutions
y(x) of (1.1) are 0(x ~2^f> ^ = o(x *a ) as x - Since a > 2,

all solutions of (1.1) are, therefore, L^O,^) and we have the limit-circle
case. Thus the result of this section is:

F. Let a > 2(3-2 and let p(t) < 0 for all t. Also, let p(t) be
Infinitely differentiable. Then (1. 2) makes (1.1) limit-circle .

-22-

8. The case a > 2(3 - 2, p(t) <0 for all t, and p(t) taking the
value zero

We do not have a complete analysis of this case but we can say
enough to indicate that the situation is more complicated than in previous
cases in that the order of the zeros of p(t), as well as a and p,
appears to affect the limit-point, limit-circle nature of (1. 2). We shall
give the discussion for the particular potential

q(x) = - x° sin2n(xli) , (8.1)

where n is a positive integer, but the ideas require only obvious
modifications for suitable more general potentials (1.2). However, it
does remain an open question to what extent (8.1) is typical of all
potentials (1. 2) falling under the heading of this section. There are
certainly complications if p(t) has an infinity of zeros of order 2n in
(0, a), or more generally if it vanishes at a point which is* not a zero of
a specific order.

We obtain first a limit-point result for (8.1). We take

, .1/p -1/2 . . , ,1/p -1/2 . ,, ,w

a = mrr) r - m and b = (rmr) + m in 2.2) (or in

m m

Corollary 1 of [ 5)). Then (8.1) will be limit-point if

xa sin2n(x^) < Cm (8.2)

in (a , b ), where C is a constant. To ensure that the (a , b )
mm’ m m

are non-overlapping, at any rate when m is large enough, we take

P < 2 . (8.3)

Since x = (mir)

+ 0(m ) in (a , b ), (8.2) is satisfied if

m m

a 11.

i.e. if

a + (n - l)p < 2n .

This condition implies (8. 3) since a > 2. Hence we have

G . Let a + (n - 1)(3 < 2n. Then (8.1) makes (l.J) limit-point-
We now make a conjecture.

H (conjectured). Let a + (n - l)p > 2n. Then (8.1) makes (1.1)
limit-circle.

We support this conjecture with the following remarks. Considering
(3.2), we seek an approximation to solutions of

z(t) + {bt + (3 ^t^& sin2nt}z(t.) = 0 (8.4)

throughout an interval [ (m - ~)n, (m + ~)ir), where m is a large
integer and, as in § §6 and 7, 6 = > 0. We define

P(t) = bt + (3 t sin t ,

i(t) = (n + l)1/(n+i)( / P^'(u)du)i/(n+1) , (8. 5)

rmr

and

f(t) = e^n(t)p"^(t) .

Then it can be shown that, both when t - mir is small and when t - mir
is exactly of order 1,

V

-24-

. . .

ft

(a) f(t) X m'^/(nfl) ,

(b) f(t)W(4) satisfies an equation approximating to (8 . 4), where
W(4) is a solution of

(8.6)

^ + eZnw = 0 .

We omit the routine details of the calculations. Now (b) suggests that
solutions z(t) of (8.4) are approximately of the form f(t)W(4) and
hence, by (3.1), that (1.1) is limit-circle if

Yj I |f(t)W(£) |V2+2/Pdt < » .

m (m--)-*

By (8. 5) and (8. 6),

4 P2(t) = f (t) X m

6/(n+l)

Hence (8. 8) holds if

X m-*V(n+l)-3+2/p |w(5)|2d£ < »,

m 4{(m--|)ir}

where we have used (8.6) again. Since all solutions W(4) of (8.7) are
0(4 ?°) as 4 - °° , the integral term here is bounded if n > 1. (The
case n = 1 introduces a negligible logarithm.) Hence (8.8) holds if

n + 1 (3

Since 26 = -2y = -2 + (a + 2)/p, this reduces to a + (n - 1)(3 > 2n
as required.

The rigorization of this argument would appear to involve some
complicated analysis on the lines of the Langer-Titchmarsh approach to
turning points. Although it is hoped that a treatment of this will appear
in due course, the details have not been carried through at the present
time.

-26-

a = 2p - 2

In this section, it is convenient to consider, in place of (1.1),

the equation

y"(x) - |q(x) + bp2x 2}y(x) = 0 ,

where b is as in (3.2). Since the coefficients of y(x) in (9.1) and
(1.1) differ only by a term bp2x 2 which is bounded in the neighborhood
of x = «, the limit-point, limit-circle nature of (1.1) is the same as
that of (9 . 1) at x - <* (cf. [4, p. 225)). When the transformation ( 3. 1)
is applied to (9.1), we obtain, in place of (3.2), the simpler equation

z(t) - p 2t 2^p(t)z(t) = 0 .

In the present case when a - 2p - 2, v = 0 and we have the periodic

equation

z(t) - p p(t)z(t) = 0

(9.2)

The limit-point, limit-circle nature of (1.1) is connected to the stability
nature of (9 . 2), a connection which was noted by Sears [19] in a not
dissimilar context. For completeness, we give here the details of this
connection and we refer to [8, §§1.1-3] for the necessary theory of (9.2).

If (9.2) is stable, all solutions of (9.2) are bounded in (0,»)
and so, by (3.1), all solutions y(x) of (9.1) are

0(x“^"^/2) = 0(x.~a/4)

as x -* ». Since a > 2, we have the limit-circle case.

If (9. 2) is unstable, (9 . 2) has an exponentially large solution as
t * « and the corresponding y(x) is certainly not l/(0,*>). Hence
we have the limit-point case.

If (9.2) is conditionally stable, but not stable (i.e. case D2 or
case E2 of [ 8, §1.2] holds), (9 . 2) has a solution z(t) of the form

z(t) = tPj(t) + Pz(t) ,

where F^(t) and P^t) have period a or 2a. For this z(t) we have

/" lz2(t) lt'2+2/fidt = -

and hence, by (3.1), the corresponding y(x) is not L2(0,oo), Thus we
have the limit-point case again.

The result, then, of this section is:

I. Let a = 2(3-2. If (9.2) is stable, then 1 1. 2) makes (1.1)
limit-circle. Otherwise. (1.2) makes (1.1) limit-point.

An example in which both the possibilities in I are realised is
that in which (9.2) is the Mathieu equation. Here,

p(t) = - (3^(\ - q cos 2t) ,

where \ and q are constants with q * 0. Given q, (9.2) can be
made both stable and unstable by choice of \ - see, e. g. , [8, §2.5].

Acknowledgement. Work on this paper was done while we were at the
University of Dundee during the Summer Term of 1974 in connection with the
Symposium on Spectral Theory and Differential Equations, which was
supported by the Science Research Council. We thank Professor W. N. Everitt
for the invitation to visit the Mathematics Department of the University of Dundee.

-28-

References

F. V. Atkinson, On asymptotically periodic linear systems. J. Math.

Anal. Appl. 2-4 (1968), 646-6S3.

, Limit-n criteria of integral type. Proc. Roy. Soc.

Edinburgh (A) 7 3 (197S), 167-198.

1 . Br i nc k , Self-adjointness and spectra of Sturm-Liouville operators.
Math. Scand. 7 ( 1 9 S9) , 219-239.

E. A. Coddington and N. Levinson, Theory of ordinary differential
equations. (McGraw-Hill, 1 9 S 5) .

M. S. P. Eastham, On a limit-point method of Hartman. Bull. London
Math. Soc. 4 (1972), 340-344.

, Limit-circle differential expressions of the

second-order with an oscillating coefficient. Quart. J. Math. (Oxford) (2)
24 (197 3), 2S7-263.

MM

, Second- and fourth-order differential equations

with oscillatory coefficients and not of limit-point type. Proceedings
of the Scheveningen Conference on the Spectral Theory and Asymptotics
of Differential Equations (Edited by E. M. de Jager, North- Holland
Mathematics Studies, Vol. 13, 1974).

, The spectral theory of periodic differential

equations. (Scottish Academic Press, Edinburgh, 197 3).

, The limit- 3 case of self-adioint differential

expressions of the fourth order with oscillating coefficients. J. London
Math. Soc. (2) 8 (1974), 427-437.

-29-

10. M. S. P. Lastham, W. D. Lvans and J. B. McLeod, 1'he essential selt-
adiointness oi Schrodinuer-type operators. Arch. Rational Mech. Anal.

60 (1976), 185-204.

mm

11. P. Hartman, The number of L^-soluttons ot x" ♦ q(t)x - 0. Amer. J.
Math. 73 ( 1 9 Si), 635-645.

12. R. S. Ismagilov, Conditions tor self-adiointness of differential operators
of higher order. Soviet Math. 3 (1962), 279-283.

1 3. , On the self-adjointness of the Sturm-Liouville

operator, Uspehi Mat. Nauk 18 (1963), 1 61-166.

mm

14. I. Knowles, Note on a limit-point criterion. Proc. Amer. Math. Soc.

4j. ( 197 3), 1 17-1 19.

1 5. J. B. McLeod, On the spectrum of wildly oscillating functions. J. London
Math. Soc. 39 (1964), 623-634.

MM

i 6. . II. ibid. 40 ( 1966), 655-661.

17. . III. ibid. 40 ( 1965), 662-666.

MM

18. J. J. Mahony, Asymptotic results for the solutions of a certain
differential equation. I. Austral. Math. Soc. 1 3 (1972), 147-198.

mm

19. D. B. Sears, On the solutions of a linear second-order differential
equation which are of intearable square. J. London Math. Soc. 24
(1949), 207-21 5.

20. W. Walter, Differential and integral inequalities (English edition,
Springer-Verlag, 1970).

-30-

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